# Mapping Curie Depth Across Western Canada From a Wavelet Analysis of Magnetic Anomaly Data

## Abstract

In western Canada, geophysical studies infer an abrupt change in crustal temperatures between the Canadian Cordillera and the adjacent North American craton, with important implications for the tectonics and geodynamics of the area. We use a wavelet analysis of magnetic anomaly data in western Canada to map the depth to the bottom of the magnetic source, or Curie depth. This depth corresponds to the point at which crustal rocks reach their Curie temperature, thus providing an estimate of geothermal gradient. Our model is defined by a fractal distribution of magnetization characterized by the parameter *β*, as well as the depths to the top (*z*_{t}) and bottom (*z*_{b}) of the magnetized layer. Synthetic tests reveal the increased accuracy of the estimated *z*_{b} when values for *z*_{t} and *β* are fixed prior to the inversion. We set *z*_{t} to the thickness of sedimentary rocks overlying the magnetic bedrock and use various values of *β* to estimate *z*_{b}. We determine *β* a posteriori by comparing Monte Carlo simulations of predicted heat flow values (assuming a Curie temperature of 580 °C) with observed heat flow in various regions. Our results suggest a *β* value of 2.5 for the Canadian Cordillera and Slave craton, and 2 for the remaining North American craton. The Curie depths resolve geological domains and important structural features, with estimates for *z*_{b} averaging 15 ± 1 km in the Cordillera, 32 ± 3 km in the Slave craton, and 34 ± 3 km in the North American craton to the south.

## Key Points

- The wavelet transform is used to map the depth to the bottom of the magnetized source in western Canada
- We use a fractal distribution of crustal magnetization, which is seen to vary within the study area
- Curie depth estimates are 15 ± 1 km in the Cordillera, 32 ± 3 km in the Slave craton, and 34 ± 3 km in the North American craton

## Plain Language Summary

Despite a relatively high elevation, geophysical imaging of the Earth beneath Canada's western mountains reveals a distinct 150 km thinning of the tectonic plates. The plates define the rigid outer shell of the Earth, and the thickness change is coincident with a topographic difference (higher) at the surface and a temperature difference (hotter) at depth. The relationship between the high topography with the thinner tectonic plate and hotter material at depth is not well understood. The objective of this research is to estimate temperatures of the Earth's crust below the surface in western Canada using the magnetic properties of rocks. At a certain temperature, rocks lose their magnetization and therefore do not produce magnetic anomalies. By analyzing magnetic anomaly data, we can infer the depth at which this temperature is achieved, which provides estimates of the thermal gradient in the crust. Combining these estimates with heat flow measurements provides more accurate constraints on the physical properties of the tectonic plate and will allow different tectonic models to be evaluated, allowing for a better understanding of the mechanical behavior of the Canadian Cordillera.

## 1 Introduction and Tectonic Setting

The western margin of the North American continent, which today consists of the Canadian Cordillera, is composed of a patchwork of allochthonous and para-autochthonous terranes that record almost 750 million years of tectonic processes (Monger & Price, 2002). The Cordillera started to develop in the Neoproterozoic with the formation of an ocean basin on the margin of Laurentia, the newly rifted portion of the Rodinia supercontinent that eventually became the North American continent. By 545 Ma, the rift had evolved to seafloor spreading, where the thick passive margin sediments presently found in the Western Canada Sedimentary Basin were deposited (Bond & Kominz, 1984; Monger & Price, 2002). From the middle Paleozoic, the divergent intraplate setting shifted to a convergent setting, which led to a series of magmatic arcs forming the many oceanic and accretionary terranes that eventually accreted onto ancestral North America in the Mesozoic (Monger & Price, 2002, and references therein). Orogen-normal compression, crustal thickening, and uplift in the Jurassic are interpreted as an important episode of Cordilleran mountain-building (Dusel-Bacon et al., 2002; Foster et al., 1985; Hansen & Dusel-Bacon, 1998; Monger & Price, 2002; Murphy et al., 2006). The Cretaceous is characterized by right-lateral transtension in the northern Canadian Cordillera, similar to the right-lateral transtension in the Eocene across the Canadian Cordillera (Gabrielse et al., 2006; Monger & Price, 2002). In the southern Canadian Cordillera, the subduction of the Juan de Fuca plate beneath North America was responsible for the accretion of Cenozoic terranes onto the western margin of the Cordillera in the Eocene (Cook, 1995).

Cutting across the entirety of the Cordillera is a major northwest/southeast striking physiographic lineament which includes the Rocky Mountain Trench (RMT) in British Columbia and the Tintina Fault in Yukon and Alaska (Figure 1a). The crustal-scale, right-lateral Tintina Fault accommodated >400 km of displacement between the Cenozoic to the present day (Gabrielse et al., 2006; Hayward, 2015; Leonard et al., 2008; Roddick, 1967; Saltus, 2007). This marks an important boundary in terms of geology, separating the oceanic and accretionary terranes west of the fault from the deformed and weakly metamorphosed sedimentary rocks from ancestral North America's passive margin on the eastern side, up to the Cordilleran Deformation Front (e.g., Colpron et al., 2007; Gordey, 2013). East of the Deformation Front, an aggregation of multiple Archean cratons constitutes the North American craton, overlain by Proterozoic juvenile arcs and orogens to the west near the craton edge, overlapping to the other side of the Cordilleran Deformation Front (Figure 1a).

The present-day geophysical signature of the North American craton is quite different than that of the Canadian Cordillera. Many studies have defined Cordilleran crust and lithosphere as relatively hot and thin compared to the North American craton, with the average heat flow in the Cordillera being roughly twice the average heat flow in the craton (Lewis et al., 2003; Mareschal et al., 1999; Rolandone et al., 2002), and the elastic thickness in the Cordillera rapidly increasing from <10 to >70 km at the Cordillera-craton boundary (Audet et al., 2007; Flück et al., 2003). Mantle seismic velocities are lower beneath the Canadian Cordillera than beneath the adjacent craton; however, at crustal to uppermost mantle depths, the variations in seismic velocities vary according to important tectonic boundaries in the Cordillera (Bao et al., 2014; Hyndman et al., 2009; Kao et al., 2013; McLellan et al., 2018; Perry et al., 2002; Schaeffer & Lebedev, 2014). There is also an abrupt transition between the 30 to 35 km thick crust in the Cordillera and 40 to 45 km thick crust in the craton (Clowes et al., 1995, 2005; Cook et al., 2012; Fernández-Viejo et al., 2005; Kao et al., 2013; Mooney et al., 1998; Perry et al., 2002; Tarayoun et al., 2017). Such a shallow Moho, despite the high elevations in the Canadian Cordillera, can be explained by the thermal expansion of the upper mantle, corroborated by high upper mantle temperatures inferred from the thin elastic thickness, low seismic velocities, and high heat flow (Currie & Hyndman, 2006; Hyndman & Currie, 2011). The mechanisms responsible for generating and maintaining this high heat are not well constrained. Geodynamic models that focus on this problem require accurate estimates of crustal temperatures; however, in western Canada, heat flow measurements are sparse and unevenly distributed (Figure 1b; Perry et al., 2010, and references therein).

Curie depth estimates, which correspond to the depth where crustal rocks reach their Curie temperature (~580 °C for magnetite; Dunlop & Özdemir, 1997), give independent temperature constraints over an entire area where magnetic anomaly data are available. Curie temperature is defined as the temperature at which ferrimagnetic and ferromagnetic materials become paramagnetic. The thickness and depth to the base of the magnetized crust is reflected in the wavenumber content of the magnetic anomaly data, and the Curie depth can be estimated directly from the radially averaged power spectral density function using curve-fitting methods (e.g., Bouligand et al., 2009). Therefore, this approach provides more direct estimates of crustal temperatures than those obtained from elastic thickness or seismic tomography models (e.g., Kaban et al., 2014).

Since the wavenumber content of magnetic anomaly data varies spatially, and the long-wavelength anomalies require large windows that reduce the spatial resolution, spatio-spectral localization techniques are necessary. Most commonly, the Curie depth is estimated within square windows of various sizes that are moved across the study region to produce a map (e.g., Bouligand et al., 2009; Li et al., 2017; Wang & Li, 2015; Witter & Miller, 2017). The segmentation of the signal using moving windows has significant drawbacks including spectral leakage and bias. Moreover, an optimal window size is set in advance, which must be large enough to accurately characterize the long wavelengths of the signal (typically 6–10 times the depth to the bottom of the magnetic source; Ravat et al., 2007), limiting the spatial resolution. Here we use a 2-D wavelet transform, where wavelets at different azimuths and scales are convolved with the entire magnetic anomaly grid, avoiding the segmentation of the signal into finite-size windows. The wavelet transform offers a better compromise between spatial and wavenumber resolution than moving windows since the wavelet transform uses a combination of multiple wavelet scales to construct a power spectrum at each grid point (e.g., Audet et al., 2007; Kirby & Swain, 2014; Pérez-Gussinyé et al., 2009; Swain & Kirby, 2003).

The shape of the radially averaged power spectrum of magnetic anomalies can be modeled as a function of the depths to the top (*z*_{t}) and bottom (*z*_{b}) of the magnetized layer, and *β*, which describes the fractal distribution of crustal magnetization (e.g., Bansal et al., 2011; Bouligand et al., 2009; Li et al., 2017; Maus et al., 1997; Maus & Dimri, 1996; Pilkington et al., 1994; Pilkington & Todoeschuck, 1993). Many Curie depth studies assume that the distribution of magnetization in the crust is random, which is represented by *β* = 0 (Blakely, 1988; Ross et al., 2006; Tanaka et al., 1999). Previous studies have demonstrated the trade-off between the fractal parameter and *z*_{b} results, and shown that assuming a randomly magnetized crust results in greatly overestimated *z*_{b} (e.g., Bouligand et al., 2009; Witter & Miller, 2017).

In this study, we use a fractal correction method to remove the effect of the parameter *β*, followed by nonlinear curve fitting to find the best fit theoretical power spectrum in a least squares sense. We first test the inversion on synthetic data to show the strengths and limitations of the method. These tests demonstrate that *z*_{b} can be estimated with a good spatial resolution when using the wavelet transform, and *z*_{b} can be more accurately estimated if *z*_{t} and *β* are fixed to the correct value. We then apply our method to data from western Canada. Our results indicate that the Tintina Fault system, RMT, and Slave craton are well resolved by the estimated *z*_{b} map. As expected, the *z*_{b} values suggest much higher geothermal gradients in the Canadian Cordillera than in the North American craton, and the *z*_{b} results are generally negatively correlated with surface heat flow measurements.

## 2 Methodology

### 2.1 Layered Magnetization Model

*z*

_{t}and

*z*

_{b}. A magnetized layer of a certain depth and thickness has a predictable spectral shape; therefore, the depths to the top and bottom of the magnetic source can be inferred from the power spectrum of the magnetic anomalies using curve-fitting methods. For a magnetic layer confined between

*z*

_{t}and

*z*

_{b}, the equation from Blakely (1988) can represent the 2-D theoretical power spectral density (PSD) function of the total-field anomaly:

_{M}(

*k*

_{x},

*k*

_{y}) is the 2-D PSD of magnetization, and

**∣ is the norm of the 2-D wavevector**

*k***= (**

*k**k*

_{x},

*k*

_{y}), C

_{m}is a constant, and Θ

_{m}and Θ

_{f}are the complex functions of

*k*

_{x}and

*k*

_{y}that depend on the orientations of the dipole , and the unit vector parallel to the ambient field , respectively. The radial average of the total-field anomaly PSD is given by

*k*= ∣

**∣ and**

*k**A*is a constant that depends on the direction of magnetization and regional magnetic field.

Deep magnetic sources are characterized by longer-wavelength anomalies, as shorter-wavelength anomalies are more attenuated. Therefore, in the PSD function, the power at low wavenumbers is higher when the base of the magnetized layer is deeper. The depth to the top of the magnetized layer affects the high wavenumber end of the spectrum, which has higher power for tops at shallower depths. Consequently, areas that are magnetized at shallow depths will be characterized by short-wavelength anomalies.

### 2.2 Crustal Magnetization

*z*

_{t}and

*z*

_{b}from the observed PSD of the total-field anomaly requires some assumptions on the statistical properties of magnetization. A common assumption is that crustal magnetization is a random distribution of dipoles in the Cartesian plane (e.g., Blakely, 1988; Ross et al., 2006; Tanaka et al., 1999; Witter & Miller, 2017), in which Φ

_{M}(

*k*) is a constant and we can solve directly for

*z*

_{t}and

*z*

_{b}from the total-field anomaly spectrum. However, magnetization typically follows a fractal distribution, meaning that crustal magnetization has some degree of self-similarity (Maus & Dimri, 1995, 1996; Pilkington et al., 1994; Pilkington & Todoeschuck, 1993). The degree of self-similarity depends on a number of factors including the lithology and geological processes and is likely to vary within a study area (Bansal et al., 2010; Gettings, 2005, 2012; Leonardi & Kumpel, 1996; Maus & Dimri, 1995, 1996; Pilkington, 2007; Pilkington & Todoeschuck, 1993). Since

*z*

_{b}is estimated by analyzing the shape of the PSD of magnetic anomalies (e.g., Spector & Grant, 1970), self-similarity is an important characteristic to consider: a fractal distribution is expressed by a negative slope in the PSD of magnetic anomalies on a log-log plot affecting all wavenumbers, whereas a random distribution has a flat power spectrum. The negative slope of the PSD of magnetization on a log-log scale is given by

*β*:

The negative slope of the power spectrum decreases with increasing degree of randomness, and *β* = 0 represents a random function whose PSD is flat. A fractal model for Φ_{M} in equation 3 is introduced by substituting Φ_{M} by *k*^{−β}.

### 2.3 Wavelet Analysis

*ψ*, at scales

*a*and azimuths

*θ*(Foufoula-Georgiou & Kumar, 1994). The different scales and azimuths allow for the signal to be analyzed at different bandwidths and directions, and they are translated to analyze the signal at different locations (

**b**):

**r**is the location on the 2-D physical plane (r

_{x}, r

_{y}) and

**C**

_{θ}is the rotation operator

The scaled, rotated, and translated wavelets allow for the characterization of a nonstationary signal in spectral space.

*f*(

**r**) and is the complex conjugate of the Fourier transform of .

**k**

_{0}is the wavevector that the Morlet wavelet is centered on in the wavenumber domain. This wavevector sets the spectral resolution and must possess a magnitude

*k*

_{0}= |

**k**

_{0}| higher than 5.336 to satisfy the zero-mean condition. Parameter

*k*

_{0}is inversely proportional to the bandwidth of the wavelet; therefore, a high

*k*

_{0}has better resolution in spectral space and lower spatial resolution than a low

*k*

_{0}. High

*k*

_{0}values will therefore produce PSD estimates that are closer to Fourier-based estimates and will result in smoother maps of estimated parameters. The relation between the wavelet scale and equivalent Fourier wavenumber

*k*

_{F}is given by .

**A**is the 2 × 2 anisotropic diagonal matrix, where the first and second elements are set to 1 to create an isotropic wavelet in this case (Antoine et al., 1993, 1996; Kumar, 1995).

*∆θ*is the azimuthal extent. The parameter p is the amplitude at which the Morlet wavelets intersect with the adjacent superimposed wavelets and affects the number of Morlet wavelets used. We use p = 0.75 to obtain an optimal fan wavelet that has constant amplitude in all azimuths (see Kirby, 2005). Instead of averaging the individual wavelets over all azimuths, we calculate the wavelet coefficients for all scales and azimuths (equation 6) and average the wavelet power spectrum:

The fast Fourier transform is employed when transforming between the space domain and spectral domain to decrease the processing time, using the magnetic anomaly grid padded with zeros to the next power of 2 in each direction. For a more in-depth description of the wavelet transform applied to potential field data, see Kirby (2005) and Audet and Mareschal (2007).

### 2.4 Application to Magnetic Anomaly Data

**k**|

^{β}. To simplify the analysis, we calculate the logarithm of the radial average of the local PSD (i.e., logarithm of equation 11), to compare with the theoretical PSD for a 2-D magnetized layer (i.e., logarithm of equation 3):

The best fit curve, and thus the best fit *z*_{t} and *z*_{b}, is then found using nonlinear least squares (Bouligand et al., 2009), and the goodness of fit is calculated using the reduced chi-square statistic. Bouligand et al. (2009) also used a 3-D model for crustal magnetization with a corresponding *β*_{3D}, whereas a 2-D magnetization model is used here, so the *β*_{3D} value used in Bouligand et al. (2009) is equivalent to a value of *β*+1 in this study (Bouligand et al., 2009; Li et al., 2013; Maus et al., 1997). Bouligand et al. (2009) demonstrated that estimating *z*_{t}, *z*_{b}, and *β* jointly may lead to inaccurate results; therefore, in this study, we estimate depth parameters with *β* fixed to a constant value for the entire study area. Multiple maps of *z*_{t} and *z*_{b} created using a different *β* value from 2.0 to 3.5 are shown, but inversions were also carried out with *β* values from 1.5 to 4.0, which are typical values estimated in crustal rocks (e.g., Maus et al., 1997; Maus & Dimri, 1995; Pilkington & Todoeschuck, 1993).

## 3 Synthetic Examples

The synthetic tests in the present study build on the work of Bouligand et al. (2009) by testing the inversion when both *β* and *z*_{t} are fixed, which shows that *z*_{b} values are more accurate when it is the only unknown parameter. Moreover, we go further by testing the inversion on a magnetized layer that has a variable thickness, showing that varying *z*_{t} and *z*_{b} in the synthetic models results in less reliable *z*_{t} and *z*_{b} estimates than a uniform magnetic layer. We test the inversion by generating synthetic magnetic anomaly maps using a set of parameters that describe the magnetization: *z*_{t}, *z*_{b}, and *β*. First, the Fourier spectrum of magnetization is generated by creating a grid of wavenumbers with random phase; the wavenumber grid is then raised to the exponent − *β*/2, and inverse Fourier transformed. To obtain the magnetic anomaly map, we multiply the Fourier transform of the magnetization by the function H in equation 2 using the specified values for *z*_{t} and *z*_{b}. In these examples, we only test the recovery of *z*_{t} and *z*_{b} assuming a known *β* parameter. The synthetic map is therefore corrected for *β* using the input value. We then calculate the wavelet transform and wavelet power spectra using the parameter *k*_{0} = 10, and apply curve fitting at each point of the 2-D map using equation 11 to estimate *z*_{t} and *z*_{b}. This step is carried out 100 times using the same model parameters but a different random seed generator in order to obtain the statistics of the recovered parameters; we thus obtain 100 *z*_{b} maps (and 100 *z*_{t} maps when *z*_{t} is not fixed to a set value), and we calculate the average and the standard deviation. This illustrates how accurately and how consistently the inversion reproduces the depth and thickness of the magnetized layer. For each synthetic model, we also set *z*_{t} to its correct (i.e., input) value and apply curve fitting with a single unknown parameter to estimate *z*_{b}. Selected results are shown in Figures S1, 2, and 3. Figure S1 shows the inversion results for a magnetized slab with uniform *z*_{t} and *z*_{b}, and illustrates the trade-off between *z*_{t} and *z*_{b}. These examples show how setting *z*_{t} to the correct (i.e., input) value provides much more accurate *z*_{b} results.

*∆T*and

*z*

_{0}= . In the case of a magnetic layer of variable thickness, the input

*z*

_{t}and

*z*

_{b}vary laterally; therefore, the average and the standard deviation are calculated at each grid point in the map from the 100

*z*

_{t}and

*z*

_{b}maps generated.

Figures 2a and 2b show the depth to the top and bottom of the magnetic layer, respectively, and Figures 2c and 2d show the curve-fitting results for a magnetic layer with a uniform *z*_{t} and variable *z*_{b}. The results for *z*_{b} when *z*_{t} and *z*_{b} are jointly estimated have relatively large standard deviations, and the estimated *z*_{b} are not accurate near the transitions from deep to shallow *z*_{b} (Figure 2c). Where there is a transition from deep to shallow *z*_{b}, the average estimated *z*_{b} becomes even deeper instead of transitioning to a shallower *z*_{b}. As for the average *z*_{t} map (Figure 2c), the average depth becomes shallower where *z*_{b} becomes shallower. The *z*_{t} estimates are more accurate and have a smaller standard deviation compared to *z*_{b}. When *z*_{t} is fixed to the correct value, *z*_{b} estimates are more accurate and the standard deviation is smaller relative to the results when *z*_{t} and *z*_{b} are estimated jointly; however, the estimated *z*_{b} are not able to replicate sharp transitions (Figure 2d).

Figures 2e–2h show the results when *z*_{t} varies, and the base of the layer is at a uniform depth. When *z*_{t} and *z*_{b} are estimated jointly, the correct topography in *z*_{t} is not recovered; *z*_{t} is relatively uniform but at the mean depth of the input *z*_{t} (Figure 2g). The estimated *z*_{b} is also relatively uniform, but deeper than the input value (Figure 2g). If *z*_{b} is estimated while *z*_{t} is fixed to the correct values, the estimated *z*_{b} results are still not accurate, having the inverse of the *z*_{t} topography but with amplitudes ~2 times higher (Figure 2h). In summary, estimated *z*_{t} will inevitably be biased in large lateral *z*_{t} gradients.

Figure 3 shows inversion results where *z*_{t} strongly varies laterally, as expected for thick sedimentary basins that are only weakly magnetized. Here the input *z*_{t} is fixed to the depth to basement according to SEDMAP (Figure S2; Laske & Masters, 1997), and we further imposed a circular region in the center where input *z*_{b} is shallower (15 km) than the surrounding *z*_{b} (20 km) to assess if a variation in *z*_{b} of this magnitude can be recovered using the wavelet transform approach, when *z*_{t} varies in a way that is realistic in western Canada (Figures 3a and 3b). The data gaps in the magnetic anomaly grid (white areas in Figure 1a) are also included in this synthetic test, by setting the magnetic anomaly values in these areas to 0 nT, to visualize the effect of these gaps on the results. As expected, the results are superior with *z*_{t} fixed to the depth to the basement during the inversion (Figures 3c and 3d). In Figures 3c and 3d, *z*_{b} gradually becomes shallower than 20 km toward the edges of the maps, starting at 200 km from the east and west edges, and 100 km from the north and south edges. The areas surrounding the data gaps are less accurate when *z*_{t} and *z*_{b} are estimated jointly, where some of the surrounding *z*_{b} values are deeper than 36 km and these areas extend less than 50 km outward from the data gaps (Figure 3c). When *z*_{b} only is estimated, the *z*_{b} surrounding the data gaps are also inaccurate but less so than when *z*_{t} and *z*_{b} are estimated jointly, and again, these areas extend less than 50 km outward from the gaps (Figure 3d). In either cases, the *z*_{b} results are less accurate where there is a sharp transition in *z*_{t}, for example, in the Bowser Basin in northern British Columbia, and at the Cordilleran Deformation Front (Figures 3c and 3d).

Based on these tests, we recommend fixing *z*_{t} to a reasonable value based on prior geological knowledge, and estimate *z*_{b} independently. However, we caution that areas with large lateral gradients in *z*_{t} will return biased *z*_{b} estimates, which is an additional source of error that is difficult to quantify.

## 4 Application to Western Canada

### 4.1 Magnetic Anomaly Data Set

The magnetic anomaly map (Figure 1a) is a portion of the aeromagnetic compilation from 1-km NAMAG data, continued to an elevation of 305 m (Bankey et al., 2002), which includes surveys from different sources such as the Geological Survey of Canada and industry. Merging grids with different specifications may have resulted in offsets or changes in the spectral content within the survey area; thus, the long-wavelength anomalies in the original merged data set are not accurately represented. The anomalies with wavelengths greater than 500 km are already filtered out of the compilation grid and replaced by downward continued satellite data, a procedure that can introduce noise in the data. For this study, the magnetic anomaly grid is decimated to 3 km and projected onto a rectangular grid. The sedimentary data are extracted from the SEDMAP model (Laske & Masters, 1997), defined on a 1° grid, which we project and interpolate (using bicubic interpolation) onto a rectangular grid with a sampling interval of 3 km (Figure 3a).

The change in spectral content of the magnetic anomalies throughout the study is visible from the magnetic anomaly map (Figure 1a). The Cordillera west of the Tintina-RMT boundary is characterized by short-wavelength anomalies, and east of this boundary until the Cordilleran Deformation Front there are little to no magnetic anomalies (Figure 1a). West of the Tintina-RMT boundary are a collection of accreted oceanic and arc terranes, and sandwiched between the Tintina-RMT and the Deformation Front are passive margin rocks of ancestral North America (e.g., Colpron et al., 2007; Gordey, 2013). The lack of magnetic anomalies in these areas may be explained by the thick sedimentary material. Arc material is often relatively highly magnetic, whereas sedimentary rocks are expected to be paramagnetic (weakly magnetic). East of the Cordillera Deformation Front, the long-wavelength anomalies generally have much greater amplitude than the anomalies in the Cordillera (Figure 1a).

### 4.2 Curie Depth

The results presented in this section are shown as maps of the depths to the bottom of the magnetic layer relative to topography, which give an independent constraint for crustal temperatures. For joint inversions of *z*_{t} and *z*_{b}, the *z*_{t} map is also shown; otherwise, the *z*_{t} is fixed to the depth to the magnetic basement shown in Figure S2.

The synthetic tests outlined above indicate that estimating *z*_{t} and *z*_{b} jointly leads to results that are less accurate. An inversion is done by jointly estimating *z*_{t} and *z*_{b}, to compare with the *z*_{b} results when *z*_{t} is fixed to a certain depth. An example is shown in Figure S3. In this inversion, *β* is set to 2.5. The saturation of the scale bar at 50 km is based on the assumption that the mantle is nonmagnetic (e.g., Wasilewski et al., 1979; Wasilewski & Mayhew, 1992); therefore, Curie depths beyond this are deemed unreasonable (the maximum depth to the Moho in western Canada is ~40 km; Cook et al., 2012; Kao et al., 2013). Some of the *z*_{b} values in the northern Canadian Cordillera are >50 km, including some reaching over 1,000 km. The mean *z*_{b} in the Cordillera is calculated from within the western box outlined in Figure 1a to avoid the areas with unreasonably deep *z*_{b} values beyond 50 km. The *z*_{b} values in the Cordillera in Figure S3 are 24 ± 4 km on average. In the Slave craton, taken from the top right box in Figure 1a, the mean *z*_{b} is 29 ± 5 and 14 ± 4 km in the rest of the North American craton.

In the following inversions, *β* and *z*_{t} are fixed to limit the number of unknown variables, leaving only *z*_{b} to be estimated. The depth to the top of the magnetic layer is chosen to be the thickness of the sedimentary layer overlying the bedrock, since this layer is most likely weakly magnetic and is assumed to not contribute to the magnetic anomalies. However, this map has limitations; for instance, the seismic model used does not resolve older, denser, and deformed sedimentary rocks such as the late Proterozoic to Devonian Selwyn Basin and Mackenzie Platform in Yukon (Gordey, 2013, and references therein).

Since the fractal parameter of magnetization for the crust in western Canada is not known, the inversion is performed multiple times with different values for *β*, ranging from 2.0 to 3.5. Figures 4a and 4b show the inversion results for different *k*_{0} and *β* values: when *β* increases, there is an important decrease in *z*_{b}. Increasing the wavelet parameter *k*_{0} has a smoothing effect on the results since wavelets with higher *k*_{0} values are wider in space. With a higher *k*_{0}, the short-wavelength patterns in the *z*_{b} maps are less affected by the short-wavelength patterns of the magnetic anomalies (Figures 1a, 4a, and 4b). As the *β* parameter increases from 2.0 to 3.5, the mean *z*_{b} value in the Cordillera decreases from 37.6 to 1.7 km. The mean *z*_{b} within the Slave Craton ranges from 5.1 to 75.8 km (with decreasing *β*), and in the Canadian Shield south of the Slave craton, the mean *z*_{b} ranges from 2.6 to 34.2 km. These mean values are calculated from the maps obtained with a *k*_{0} value of 20. Since changes in *z*_{t} are very rapid in certain areas mentioned previously such as the Bowser Basin and at the Cordilleran Deformation Front (Figure 4b), the depth to the bottom of the magnetic source in the areas is inaccurate (Figures 2h and 3d). The *z*_{b} in these areas is often with *z*_{b} = 0 km (Figure 4b), so these data points are not included when calculating the averages.

Maps with the uncertainty and reduced chi-square of the fitted parameters are shown in Figures 4c and 4d, respectively, for each of the inversions with a *k*_{0} of 20. Within each map in Figure 4c, the estimated *z*_{b} uncertainty is high where the *z*_{b} values are deeper than 50 km, and decreases with increasing *β*. The reduced chi-square value in the Cordillera and Slave craton remains relatively constant in the different inversions, but increases with increasing *β* in Canadian Shield south of the Slave craton (Figure 4d). For all *β* values, the reduced chi-square is highest where *z*_{b} is deeper than 50 km. The fit in different areas is also illustrated in Figure 5, which shows the calculated and best fit theoretical PSD from a single grid point in different areas when *β* = 2.5 and *k*_{0} = 20: the fit is superior in areas of shallow *z*_{b}. For the spectrum in Figure 5a, located in the southern Canadian Cordillera (grid point A in Figure 4b), *z*_{t} is set to 0.044 km and the best fit *z*_{b} is 13 ± 1 km, with a reduced chi-square of 1.7. The PSD curves in Figures 5b and 5c are from points located where the *z*_{b} results are aberrant. For grid point B, *z*_{t} is set to 0.028 km and the best fit *z*_{b} is 48 ± 5 km, with a reduced chi-square of 15. The fit for grid point C is worse; the best fit *z*_{b} is 82 ± 11 km, with a reduced chi-square of 20. Here the *z*_{t} is set to 1.20 km. Grid point D is located within the North American craton where *z*_{b} values are well below 50 km (Figure 4b). At point D, *z*_{t} is set to 0.53 km and the best fit *z*_{b} is 15 ± 1 km, with a reduced chi-square of 2.1.

### 4.3 Monte Carlo Simulation

The previous section compares the results for the different *z*_{b} maps of western Canada resulting from inversions with different *β* parameters. In order to find the most appropriate *β* between 2.0 and 3.5 for different regions within the study area (assuming that *β* is constant within these regions), we assume that *z*_{b} within the boxes outlined in Figure 1a correspond to the Curie depth and compare observed heat flow estimates with heat flow calculated by assuming a conductive geotherm constrained by the average Curie depth within each box. The heat flow for the Slave craton is calculated separately from the heat flow of the Canadian Shield to the south because of the distinct *z*_{b} results in the Slave craton, which may be due to a different *β* parameter.

*q*is the heat flux,

*κ*is the average bulk conductivity,

*κ*

_{n}is the thermal conductivity at 0 °C,

*T*

_{C}is the Curie temperature, and the temperature coefficient for the upper crust is

*c*= 0.001 (Lewis et al., 2003).

A range of values centered around the average *z*_{b} for each region is also used. The parameters used for the Monte Carlo simulation are found in Table S1. The average measured surface heat flow is taken from Lewis et al. (2003) for the Cordillera (92 ± 24 mW/m^{2}) and the Slave craton (53 ± 10 mW/m^{2}; Lewis et al., 2003; Russell et al., 2001; Russell & Kopylova, 1999) and 50 ± 7 mW/m^{2} is used for the North American craton south of the Slave craton (Majorowicz, 2018). Figure 6 shows lognormal histograms indicating the frequency of the heat flow values obtained from each simulation, their mean, and standard deviations, with measured surface heat flow values superimposed. For both the Cordillera and the Slave craton, the calculated mean heat flow is centered around the measured mean heat flow for *β* = 2.5 (Figure 6). Using a *β* of 2.5, the average *z*_{b} is 15 ± 1 km in the Cordillera and 32 ± 3 km in the Slave craton (in the rectangular areas outlined in Figure 1a). For the rest of the Canadian Shield, the calculated mean heat flow is closest to the measured mean heat flow if *β* = 2.0. The average *z*_{b} in this area is 34 ± 3 km if *β* = 2.0.

## 5 Discussion

### 5.1 Comparison With Geological and Magnetic Features

When *z*_{t} and *z*_{b} are estimated jointly, the only *z*_{b} feature that coincides with major tectonic or geological boundaries is the Slave craton, where *z*_{b} is deeper than the Canadian Shield to the south (Figure 4b). Based on the results of the Monte Carlo simulation, our preferred results (where only *z*_{b} is estimated) for the Cordillera and Slave craton are those with a *β* of 2.5 while the preferred results in the rest of the Canadian Shield are the results with *β* = 2.0 (Figure 4b). The higher *k*_{0} is used in the preferred model, as it smooths the small-scale *z*_{b} features that tend to follow the short-wavelength magnetic anomalies, and has a better resolution (i.e., narrower shape) in the wavenumber domain.

In northern British Columbia, Alberta, and across the Northwest Territories, the Great Slave Lake Shear Zone and Talston Arc are very visible in the magnetic anomaly map, and this is reflected in the *z*_{b} maps (Figures 1a and 7). These structures coincide with the southern and eastern extent of the Slave craton, which has distinct *z*_{b} values that are deeper than the rest of the Canadian Shield to the south: when *β* is set to 2.5, the average *z*_{b} in the Slave craton is 31.8 km, compared to 16.1 km in the rest of the Canadian Shield. If *β* is constant throughout the study area, the deeper *z*_{b} imply that the crust is much cooler than the average crust in the North American craton; however, this does not reflect the heat flow data (Lewis et al., 2003; Majorowicz, 2018; Majorowicz & Weides, 2015; Rolandone et al., 2002). Alternatively, one could interpret the shallow *z*_{b} in the Canadian Shield south of the Slave craton as a lithological contact that represents the base of the magnetized crust that is shallower than the Curie depth (Blakely, 1988); however, a contact at this depth that is consistent throughout the North American craton south of the Slave craton is unlikely. A more reasonable explanation is that there is a strong change in lithology and geological history resulting in a different *β* value for the Slave craton, and this is consistent with the heat flow simulations (Figure 6).

Perhaps the most striking feature of the *z*_{b} maps (Figures 4a and 4b) is the sharp transition between shallow (~15 km) and erroneous deep (>50 km) *z*_{b} values which coincides with the Tintina Fault in Yukon and the RMT in British Columbia. In the northern Canadian Cordillera, the region of *z*_{b} greater than 50 km continues northward and eastward until the eastern edge of the Slave craton in the Northwest Territories. In the southern Canadian Cordillera, the area of deep *z*_{b} values extends eastward until the Cordilleran Deformation Front. Rayleigh and *S* wave tomography, magnetotelluric and geochemical studies, and seismic reflection profiles have demonstrated that the Tintina-RMT boundary is a crustal-scale structural feature that defines a major change in the geology (Abraham et al., 2001; Cook et al., 2004; Ledo et al., 2002; McLellan et al., 2018; Snyder et al., 2005), and this is also evident from our results. We interpret that the deep *z*_{b} (>50 km) between the Tintina-RMT structure and the Deformation Front are due to a lack of ferrimagnetic material (i.e., the base of the weakly magnetic sedimentary or metasedimentary rocks in this area is deeper than the Curie depth). The difference in the anomalies in these areas is not due to a change in survey specifications since the locations and shapes of the different surveys do not coincide with the areas of deep *z*_{b}. Moreover, these *z*_{b} values are deeper than the Moho (Clowes et al., 1995, 2005; Cook et al., 2012; Fernández-Viejo et al., 2005; Kao et al., 2013; Mooney et al., 1998; Perry et al., 2002; Tarayoun et al., 2017), implying a magnetized uppermost mantle. Most early studies considered the mantle to be nonmagnetic (e.g., Wasilewski et al., 1979; Wasilewski & Mayhew, 1992); however, recent studies (e.g., Ferré et al., 2014) argue for the occurrence of a magnetized upper mantle in certain geological contexts. Here we make the approximation that, if the uppermost mantle is magnetized, its susceptibility is likely much lower than that for crustal rocks, and therefore, we consider a maximum depth of 50 km. In our study, however, many *z*_{b} values are >100 km in these areas, which implies a geothermal gradient of less than 5.8 °C/km (if a Curie temperature of 580 °C is assumed), which is much lower than the current estimates for this area (Clowes et al., 2005; Lewis et al., 2003; Majorowicz, 2018). In addition, the fit of the *z*_{b} in these areas is much worse than in the areas where *z*_{b} <50 km (Figures 4d and 5). Therefore, these aberrant *z*_{b} values are likely erroneous. The fit is particularly poor at high wavenumbers where the slope of the observed PSD is steeper than the one predicted using the sedimentary layer thickness for *z*_{t} (Figures 5b and 5c). This suggests that the correct *z*_{t} in these areas is much deeper than the values in SEDMAP, which corroborates the interpretation that the aberrant *z*_{b} values are in areas with very thick sedimentary material. The high reduced chi-square values therefore indicate that the model is inappropriate in this region, and *z*_{b} cannot be obtained reliably.

The synthetic tests show that *z*_{b} is not well resolved when there are strong variations in *z*_{t} (Figures 2 and 3). Most of the depths in the sediment thickness map are close to 1 km; however, in some areas, the *z*_{t} sharply increases to values above 10 km. This results in *z*_{b} values that are shallower than *z*_{t} for example in the Bowser Basin in northeastern British Columbia and in the Beaufort Sea, as illustrated in Figures 4a, 4b, and S2. The *z*_{b} values in these areas are an artifact of the inversion and cannot be interpreted as the Curie depth.

### 5.2 Comparison With Other Studies

The study area of Witter and Miller (2017) is located within the limits of our study, so a comparison between the *z*_{b} results can be made. The Witter and Miller (2017) study illustrates some of the disadvantages of using moving windows, such as the need for a buffer zone around each area that lacks magnetic anomaly data. Unlike the model presented in this paper, in Witter and Miller (2017), the Tintina Fault is not a boundary in Curie depths, which may be due to the large windows used (200 × 200 km^{2} and 300 × 300 km^{2}), some of which cover areas on both sides of the Tintina Fault characterized by a change in wavenumber content (Figures 1a and 5). Since the wavelet transform has a better spatial resolution, the Tintina-RMT boundary is resolved in the *z*_{b} maps presented herein. Similar to the results presented here, there are very deep (>50 km) *z*_{b} values east of the Tintina Fault and in northwestern Yukon in the Witter and Miller (2017) model. There is a small area of shallow *z*_{b} on the eastern side of the Tintina Fault in central Yukon which may be explained by a piece of the Slide Mountain and Yukon Tanana terranes on the other side of the Tintina Fault (e.g., Colpron et al., 2007). An area of shallow *z*_{b} appears exactly where the terrane is located (closer to the Yukon-British Columbia border) in the *z*_{b} maps presented in this paper, visible if the saturation depth shown on the maps is decreased below 50 km. This is also observed in Li et al. (2017), a worldwide model for Curie depths. This piece of the accreted terranes is composed of similar material as the terranes on the western side of the Tintina Fault (e.g., Colpron et al., 2007). West of the fault, the *z*_{b} results from this study and Witter and Miller (2017) are qualitatively similar, with shallower depths in central Yukon and slightly greater depths in northwest Yukon.

Bao et al. (2014) calculated the Curie depth (defined as the 585 °C isotherm in their study) of a southwest-northeast cross section through the southern Canadian Cordillera and a ~400 km portion of the North American craton east of the Deformation Front, based on a steady state conductive geotherm model using the method of Russell and Kopylova (1999). This is a two-layer model where the upper crust geotherm is higher than the lower crust because of increased heat generation. The 585 °C isotherm calculated in the Cordillera in Bao et al. (2014) is relatively constant at ~20 km, and markedly increases to ~80 km at the Cordillera-craton boundary and decreases eastward to ~40 km, which are deeper than the average *z*_{b} in the Cordillera and North American craton calculated in the present study. The discrepancy may be caused by the assumptions in the steady state conductive geotherm model, for instance, the upper versus lower crustal heat generation. Alternatively, the shallower *z*_{b} found in the present study could mean that the Curie temperature is lower than 585 °C, or the *β* parameter is not set to the correct value. However, even with a different *β* in the craton, there still would not be the steady eastward decrease in *z*_{b} on the eastern side of the Deformation Front (Figures 4a and 4b).

### 5.3 Comparison With Heat Flow Measurements

The Monte Carlo simulation assumes that the Curie temperatures of the crustal rocks in western Canada follow a normal distribution around 580 °C, the Curie temperature of magnetite (Dunlop & Özdemir, 1997). Even though the origin of magnetic anomalies in crustal rocks is principally due to magnetite and titanomagnetite, other ferrimagnetic minerals may contribute to the whole rock magnetic susceptibility such as hematite, pyrrhotite, and other iron and nickel alloys in some cases and may affect the Curie temperature of the crust (Blakely, 1995, and references therein). If the popular assumption that the Curie depth corresponds to a 580 °C isotherm is incorrect, the geothermal gradient estimates used in the Monte Carlo simulation will be biased. For example, the Curie temperature of titanomagnetite is lower than 580 °C at atmospheric pressure, and decreases with increasing Ti content (Hunt et al., 1995); therefore, the *z*_{b} would be at temperatures lower than assumed. Moreover, the Curie temperatures for hematite and pyrrhotite are 675–695 °C (Chevallier, 1951; Guillaud, 1951; Smith & Fuller, 1967) and 310–325 °C (Dekkers, 1989), respectively. Since these minerals have a Curie temperature that is significantly different than 580 °C, they are likely to affect the Curie temperature when present in the crust. Moreover, Curie temperatures have been shown to increase with pressure (Schult, 1970). Most crustal rock-forming minerals such as quartz and feldspars are weakly magnetic relative to magnetite and titanomagnetite (e.g., Brown & McEnroe, 2008; Hunt et al., 1995), so their whole-rock susceptibility is mainly due to a trace amount of the latter two minerals (Blakely, 1995, and references therein). If the distribution of Curie temperatures used in the Monte Carlo simulation does not accurately represent the distribution of Curie temperatures in western Canada, the predicted heat flow results from the simulation are not accurate, but a constant Curie temperature of 580 °C is assumed here because of the many unknown variables that may affect the Curie temperature.

The distribution of heat flow measurements is shown in Figure 1b. The heat flow measurements are relatively dense in southern British Columbia, and the measurements do not appear to be constant in this area, with values ranging from ~60 to ~100 mW/m^{2}. This range of values is consistent with the measurements reported for northern British Columbia (Figure 1b). In southwestern British Columbia on Vancouver Island, multiple heat flow measurements near 30 mW/m^{2} are anomalously low and do not agree with the *z*_{b} results. The low heat flow anomaly in southwestern British Columbia is due to the Juan de Fuca slab subducting beneath the Cordillera in this area, and *z*_{b} is shallow in this area. The heat flow in Yukon seems to be more consistent near 100 mW/m^{2}, but the only measurements are in the southeastern part of Yukon. The high heat flow in Yukon has been attributed to an increased heat generation in the upper crust resulting in higher measured surface heat flow (Lewis et al., 2003). However, the high surface heat flow values seem to be consistent with xenolith and seismic studies in Yukon (Francis et al., 2010; Shi et al., 1998), and the *z*_{b} estimates do not seem to be shallower in the northern Canadian Cordillera than the southern portion as a result of higher upper crust heat generation (Figures 4a and 4b). There is a dense group of measurements in east-central Saskatchewan and west-central Manitoba where the heat flow is consistently low at values below 50 mW/m^{2} (Figure 1b). Figure 1b shows that the heat flow in the Cordillera is generally higher than in the Canadian Shield; however, the sparse data make it difficult to see the smaller-scale variations in crustal temperature and assess how consistent these heat flow values are throughout the Cordillera and North American craton.

Figure 7a shows the correlation between heat flow and the corresponding *z*_{b} at the location of the heat flow measurement within the different boxes outlined in Figure 1a, for inversion results with *β* = 2.5. Figure 7b is the same scatterplot, but the *z*_{b} results within the Canadian Shield south of the Slave craton are the inversion results with *β* = 2.0, since these *z*_{b} results work best with published heat flow averages. There is an unsurprisingly large amount of scatter in both figures, owing to the variety of rock compositions, geological settings, and upper crustal heat generation. The heat generation in the upper crust is often higher than the lower crust because of a higher concentration of radioactive elements (e.g., Lewis et al., 2003). The results for the Cordillera consistently plot at low *z*_{b} <15 km, but exhibit a wide range in heat flow values, although these are >40 mW/m^{2}. The North American craton mostly plots at deeper *z*_{b} (especially in Figure 7b), and exhibits a wide range in heat flow values that are generally lower than the Canadian Cordillera. The Monte Carlo simulation results may be biased since the amount of upper crustal heat generation is uncertain in most areas; if the measured heat flow is significantly higher as a result of the upper crustal heat generation, the Monte Carlo simulation results may be biased toward a higher *β*, where the *z*_{b} results are shallower. This may also explain some high heat flow values where *z*_{b} is large in Figure 7. The steady state geotherm model of Russell and Kopylova (1999) accounts for the increased heat production in the upper crust, is consistent with Moho temperature estimates in the Slave craton (Lewis et al., 2003), and predicts the 580 °C isotherm to be at depths of ~75 km for the Slave craton, consistent with the inversion results using *β* < 2.5. The effect of the upper crustal heat generation on heat flow measurements is also highly variable within the study area since different rock types tend to have different concentrations of radioactive minerals, with felsic rocks having a greater concentration than mafic rocks (Hasterok & Chapman, 2011). Despite the variability in heat flow measurements, the negative correlation between heat flow and *z*_{b} is more apparent in Figure 7b than 7a, where most of the data points fit between the theoretical curves for the 1-D heat conductive transport model for thermal conductivities between 1.3 and 3.5 W/m°C.

## 6 Conclusions

This study estimates Curie depths across western Canada using a wavelet transform of magnetic anomaly data. Curie depth is estimated from nonlinear curve fitting of the power spectral density function calculated at each spatial location using a model that assumes a fractal distribution of crustal magnetization characterized by the parameter *β*, as well as the depth to the top and bottom of the magnetized layer. Tests performed on synthetic data sets with laterally varying model parameters show that the estimated *z*_{b} are not accurate if *z*_{t} is also estimated, and that *z*_{b} estimates are greatly improved by setting *z*_{t} to the correct value in the inversion. Furthermore, if the map of fixed *z*_{t} values has areas of rapid changes with high amplitudes, the *z*_{b} results calculated in these areas are greatly underestimated. We then applied our method to western Canada using the sedimentary layer thickness as estimates of *z*_{t} that remain fixed in the inversion, and further test various values of the parameter *β*.

The results provide crustal temperature constraints with good spatial coverage in western Canada, but yield unreasonably deep *z*_{b} in the northern part of the study area between the Tintina Fault and Slave craton, and in the southern part of the study area between the RMT and Deformation Front, since the base of the thick sedimentary rocks in these regions lies at depths greater than the Curie depth. Therefore, study areas with thick packages of nonmagnetic rocks and underestimating the *z*_{t} values used in the inversion might lead to large errors when estimating Curie depths. Based on heat flow simulations from the Curie depth estimates, a *β* value of 2.5 gives the most accurate results in the Cordillera and Slave craton, while a *β* value of 2.0 gives better results in the Canadian Shield south of the Slave craton. With these parameters, the average Curie depth is 15 ± 1 km in the Canadian Cordillera, 32 ± 3 km in the Slave craton, and 34 ± 3 km in the North American craton to the south. The elevated and uniform crustal temperatures in the Cordillera (west of the Tintina Fault) relative to the North American craton corroborate previous heat flow, seismic, and elastic thickness studies, providing independent temperature information throughout most of the study area.

## Acknowledgments

Funding for this project was provided by an NSERC grant to P.A. and a KEGS scholarship to E.G., and we are grateful for discussions with Mark Pilkington (NRCan) and Jean-Claude Mareschal (UQAM). The magnetic anomaly data is a subset from the North American Magnetic Anomaly Grid, available from the USGS Publication Warehouse (see Bankey et al., 2002). For the heat flow data, see Perry et al. (2010), and references therein.